Conjecture on the p-reductivity threshold for Bergman quotient modules

Let $I\subseteq\mathbb{C}[\mathbf{z}]$ be an ideal, let $\mathcal{Z}$ be its zero variety, and let $\mathcal{S}$ be the submodule obtained by taking the closure of $I$ in $B^2(\mathbb{B}^m)$. Bergman quotient threshold conjecture. The quotient module

$$B^2(\mathbb{B}^m)/\mathcal{S}$$

should be $q$-reductive for $q>\dim(\mathcal{Z}\cap\mathbb{B}^m)$. The source notes that this is verified when the intersection has dimension at most one, while the general assertion is open.